## Bounded rank was probable in 1950

Somehow I wrote that last post about bounded ranks without knowing about this paper by Mark Watkins and many other authors, which studies in great detail the variation in ranks in quadratic twists of the congruent number curve.  I’ll no doubt have more to say about this later, but I just wanted to remark on a footnote; they say they learned from Fernando Rodriguez-Villegas that Neron wrote in 1950:

On ignore s’il existe pour toutes les cubiques rationnelles, appartenant a un corps donné une borne absolute du rang. L’existence de cette borne est cependant considérée comme probable.

So when I said the conventional wisdom is shifting from “unbounded rank” towards “bounded rank,” I didn’t tell the whole story — maybe the conventional wisdom started at “bounded rank” and is now shifting back!

## Do all curves over finite fields have covers with a sqrt(q) eigenvalue?

On my recent visit to Illinois, my colleage Nathan Dunfield (now blogging!) explained to me the following interesting open question, whose answer is supposed to be “yes”:

Q1: Let f be a pseudo-Anosov mapping class on a Riemann surface Sigma of genus at least 2, and M_f the mapping cylinder obtained by gluing the two ends of Sigma x interval together by means of f.  Then M_f is a hyperbolic 3-manifold with first Betti number 1.  Is there a finite cover M of M_f with b_1(M) > 1?

You might think of this as (a special case of) a sort of “relative virtual positive Betti number conjecture.”  The usual vpBnC says that a 3-manifold has a finite cover with positive Betti number; this says that when your manifold starts life with Betti number 1, you can get “extra” first homology by passing to a cover.

Of course, when I see “3-manifold fibered over the circle” I whip out a time-worn analogy and think “algebraic curve over a finite field.”  So here’s the number theorist’s version of the above question:

Q2: Let X/F_q be an algebraic curve of genus at least 2 over a finite field.  Does X have a finite etale cover Y/F_{q^d} such that the action of Frobenius on H^1(Y,Z_ell) has an eigenvalue equal to q^{d/2}?

## Two estimation questions

It\’s apparently customary, when being interviewed for a job in the consulting industry, to be asked to estimate various numerical quantities:  how many cars are rented each week in the United States?  What proportion of the total mass of American citizens is made up of males?  I think that in asking these questions the interviewer is testing your ability to carry out rapid approximate quantitative reasoning, or, alternatively, to make confident assertions about whose truth you\’re almost completely ignorant — both important skills in that line of work.

Anyway, here are two questions.  I know the answer to the first one, and will reveal it tomorrow.  Put your unreasonably confident answers in comments!

1. What is the total number of living alumni (all degrees) of UW-Madison?
2. What is the population of the largest U.S. city without a Chinese restaurant?

Update: (24 Mar)  Commenter QXW is in the lead on question 2, observing that the city of Rye, NY (pop. 14,955) has no Chinese restaurants per Google Maps.  Can it really be true?